Does the axiom of non-reflexivity follow from the definition of transitivity?

In the 48th CIDOC CRM and 41st FRBR CRM sig meeting (virtual), uUpon discussing issue 406 (transitivity and quantification of properties), MD suggested that the axiom of non-reflexivity be introduced for all properties that have part-of semantics. 
CEO proposed that the non-reflexivity requirement be examined separately –as part of the semantics imposed by a many-to-many quantification. HW CEO to produce a text that explains the semantics of transitivity vs non-reflexivity in the context of discussing P73.

October 2020

In the 50th joint meeting of the CIDOC CRM SIG and SO/TC46/SC4/WG9; 43nd FRBR – CIDOC CRM Harmonization meeting, CEO shared some background information on this issue (MD had suggested that all transitive relations should be non-reflexive, CEO had objected to that) and presented his HW. 

CRM properties are binary relations over classes. The possible relations are: transitivity, symmetry, asymmetry, reflexivity, irreflexivity. Binary relations (i.e. Ps in the CRM universe) correspond to partial orderings, i.e. they do not hold for every pair of elements. Reflexivity depends on whether the partial ordering has been defined as strict (< hence non-reflexive) or non-strict ( ≤ hence reflexive). Asymmetry implies irreflexivity.  

Whether transitive relations are also irreflexive in the CRM, stems from whether Ps are defined as strict partial orderings –which they are. Hence, the axiom of non-reflexivity does not follow from the axiom of transitivity, but from the fact that the CRM properties are construed as strict partial orderings. 
We need to add an axiom for asymmetry to the transitive properties, as this practice would implicate that they are also irreflexive.

Specifically, for P198: we either define it to be irreflexive or asymmetric –this is missing information for the time being, which allows a non-strict interpretation of the property. 

Proposal: To adjust all the relevant scope notes and add the necessary FOL axioms (for P198, add asymmetry in the list of axioms).
 
Decision: proceed as indicated below 
HW: CEO check all the transitive properties for asymmetry/non-reflexivity and report on the next meeting –share list with MD and then go over it at the SIG. 
HW: MD to reformulate the definitions of axioms in English, and then SdS will proofread. The resulting text to be incorporated in (a) the glossary, (b) the scope note guidelines, (c) and where necessary into the individual scope notes. 

June 2021
 

Post by Martin, 3 October 2021

Dear All,

I have extended the description of reflexivity with a motivation for normal users, and created a clone description of non-reflexivity, in yellow below. "Asymmetry" should disappear in the CRM following past decisions.

reflexivity
    

Reflexivity is defined in the standard way found in mathematics or logic: A property P is reflexive if the domain and range are the same class and for all instances x, of this class the following is the case: x is related by P to itself. The intention of a property as described in the scope note will decide whether a property is reflexive or not. An example of a reflexive property is E53 Place. P89 falls within (contains): E53 Place. Since geometric areas can be arbitrarily close to each other, the distinction, if two places with unprecisely known extent  are identical or are contained one in the other, can be difficult or unknown. Defining this property as reflexive allows for describing in one statement the topological constraint that a place x is either contained in a place y or identical to y. However, it is not meant to instantiate this property in a knowledge base for all instances of the domain class. In First Order Logic, we denote reflexivity by:

  “Pnn(x,x)”          

Non-reflexivity
    

Non-reflexivity is defined in the standard way found in mathematics or logic: A property P is non-reflexive if the domain and range are the same class but for all instances x, of this class the following is the case: x cannot be related by P to itself. The intention of a property as described in the scope note will decide whether a property is non-reflexive or not. An example of a non-reflexive property is E18 Physical Thing. P46 is composed of (forms part of): E18 Physical Thing. Since instances of E18 Physical Thing are required to be distinct, it is reasonable to use the property P46 is composed of only for associating an instances of E18 Physical Thing with a part being different from the whole. In logic, this is expressed by non-reflexivity. In First Order Logic, we denote non-reflexivity by:

    “ ¬Pnn(x,x)”

Post by Christian-Emil, 5 October 2021

Dear all,

I checked the issue and have produced a HW document based on version 7.2. This document is not finished. It contains an introductory session about the terms and then a complete list of all properties with identical domain and range with comments. The conclusion is that the issue is technical and requires  a clean up of the document.

Best,

Christian-Emil

In the 51st CIDOC CRM & 44th FRBRoo SIG meeting, the SIG reviewed HW by MD and CEO. 

(a) HW by MD: additional explanation for the axiom of [+/-] reflexivity in the Introduction section

Decision: Accept both definitions --the definition of reflexivity is to substitute the existing one, and non-reflexivity will be added to the terminology list. Details here. 
HW: SdS make editorial changes where necessary, make sure that reflexivity applies only if necessary (not trivially). 

(b) HW by CEO: exhaustive check on the list of properties for [+/-] transitivity, [+/-] reflexivity, [+/-] symmetry 

Decision: Postpone discussing this subtask, first give CEO a bit more time to go through the rest of the properties and then share the updated document through the SIG mailing list (by the end of October).  SIG members are invited to give feedback. The decision will ultimately be reached by an evote